Approximate Solution of Singular Optimization Problems

We study the optimal control problem for systems described by nonlinear elliptic equations. We have no information about the existence and uniqueness of the solution for some particular control. The extremum problem may be unsolvable. We regularize the problem by using a combination of the penalty method and the Tikhonov method. For the regularized problem, we prove the existence of the solution and find necessary conditions for optimality in the form of variational inequalities. We show that the regularization method used in this paper allows one to find an approximate (in some sense) solution of the original problem.